Re: [Phys-l] stranger than fiction

[John Denker <jsd@av8n.com>]

In the context of Green functions:

On 02/08/2009 07:40 PM, Stefan Jeglinski wrote:

> So if it's conventionally defined in terms of delta functions, did GG 
> then use them? 

No.

> Or, did GG use a more convoluted approach? 

That is a singularly bad pun.


========================

Green's approach was not straightforward.  Instead, he took the 
time-honored straightbackward approach:  he asserted the answer 
in the first equation on the first page, and then spent the rest 
of the chapter arguing that the answer was correct.

1) That is a viable approach.  You can use a Green function
without necessarily having a systematic understanding of where
it came from.  "Hey, look at this nifty convolution kernel I
found."

2) He made things easier on himself by inventing (also on the 
first page) the notion of "potential".  This meant he was solving 
for a scalar rather than a vector.  This is significant because 
vectors would not appear on the scene until 60 years later.

3) He got lucky because the Green function for the potential (as
a function of charge) has a particularly simple form.  It's hard 
to imagine anything much simpler than 1/|Δr|

4) He got lucky because charge uniformly distributed on a sphere
produces, at places outside the sphere, a potential provably 
identical to the one produced by charge concentrated at the center.  
That means for this application you can replace delta functions 
with little spherical charge distributions ... thereby avoiding 
singularities with little or no loss in accuracy.

============

All this explanation makes Green's work seem somewhat less mind-
boggling, but it is still pretty high up there on the bogglometer.
Sometimes you just have to take your hat off to these guys and
say that ordinary mortals could not have done what they did.

Keep in mind that Green's work was 40 years before Maxwell wrote
down the Maxwell equations.

He studied electrostatics because there was no such thing as
electrodynamics, and would not be for another 40 years or so.

Green worked in a windmill grinding flour.  At this point in his 
life, his total schooling consisted of about one year of elementary 
school.

This was not the good old days.

=============================

> Well, I've often heard you write that there is no particular reason 
> to explain a physics topic as it was done historically; in fact, more 
> modern approaches might make "more sense." 

Yeah.

In an introductory class, it would be insane to ask students to 
retrace Green's steps.

I am always astonished when teachers claim they use the "historical
approach" to organize and motivate the study of science, especially
in an introductory course.  They cannot possibly be doing what they
say they are doing.  The result is a disservice to students and an
insult to past and present generations of scientists.  For more on
this, see
  http://www.av8n.com/physics/best-evidence.htm